Major problems

This is part of an algebraic topology problem list, maintained by Mark Hovey.

  1. The biggest problem, in my opinion, is to come up with a specific vision of where homotopy theory should go, analogous to the Weil conjectures in algebraic geometry or the Ravenel conjectures in our field in the late 70s. You can't win the Fields Medal without a Fields Medal-winning problem; Deligne would not be DELIGNE without the Weil conjectures and Mike Hopkins would not be MIKE HOPKINS without the Ravenel conjectures. We can't all be Deligne or Mike, but making the conjectures requires different talents than proving them, and more of us might have a chance. This was actually my motivation for making this list; to provide a forum for conjectures so that we might collectively be able to form a program analogous to the Weil conjectures. This would make a huge difference to our field, I think. Of course, they have to be somewhat accessible conjectures, which the problems below may not be!

  2. The generating hypothesis, which asserts that the stable homotopy functor is faithful on the category of finite spectra. That is, if f is a map of finite spectra such that pi_* f is 0, then f is nullhomotopic. An unbelievable consequence of this is that the stable homotopy functor is full as well. This conjecture has withstood serious attempts for many years, so be careful! The basic reference is P. Freyd, Stable homotopy, in {\it Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965)}, 121--172, Springer, New York, 1966; MR {\bf 35} \#2280. Devinatz and Hopkins have a program to prove the generating hypothesis when the target is a sphere; see E. S. Devinatz, The generating hypothesis revisited, in {\it Stable and unstable homotopy (Toronto, ON, 1996)}, 73--92, Amer. Math. Soc., Providence, RI, ; CNO CMP 1 622 339.

  3. Find some geometric meaning for elliptic cohomology. I believe this problem may be solvable--we keep learning new things about it. One thing I will say here; if I am called to referee a paper on elliptic cohomology that does not deal with the Hopkins viewpoint on elliptic spectra, I will almost surely reject it. The time is gone when one could write papers about the Landweber-Ravenel-Stong elliptic cohomology based on the Jacobi quartic--we now understand that that is only one of many different elliptic cohomology theories, and all papers on elliptic cohomology should now accept that and deal with it. The fundamental reference here is M. J. Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, in {\it Proceedings of the International Congress of Mathematicians, Vol.\ 1, 2 (Z\"urich, 1994)}, 554--565, Birkh\"auser, Basel, 1995; MR 97i:11043. But one should also see Grojnowski's approach to equivariant elliptic cohomology--unfortunately, this does not seem to be published, but there is a preprint. Matthew Ando has also thought about this, see M. Ando, Power operations in elliptic cohomology and representations of loop groups, Trans. Amer. Math. Soc. ; CNO CMP 1 637 129. I am sure I have left something out here as well.

  4. On the same theme, find some way of doing index theory related to elliptic cohomology. This is not really algebraic topology, but would have a major impact on our field. I don't know much about this, but people who might are Ezra Getzler and Richard Melrose on the analysis side. Richard Melrose has a theory of index theory on manifolds with corners that might possibly be relevant, and Ezra has worked on index theory on certain infinite dimensional manifolds, which again might be relevant. From the algebraic topology side, I know Haynes Miller and Mike Hopkins have thought about this some.

  5. The chromatic splitting conjecture, which is considerably more complicated to state. Basically nothing is known about this, and so this one may be more accessible. Besides Hopkins and me, Nori Minami and Ethan Devinatz have both thought about this conjecture, so might be good resources. See M. Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture, in {\it The \v Cech centennial (Boston, MA, 1993)}, 225--250, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995; MR 96m:55010 .

  6. The telescope conjecture. This is one is almost certainly wrong, but we have all been waiting for 10 years now for a correct disproof of it to appear. Mahowald, Ravenel, and Shick have claimed to have a disproof for a long time, and I would guess there is at least a 99% chance that they are right. But even so, here we have a major conjecture whose disproof is going to be incomprehensible to almost everybody in the field. Compare this to the nilpotence theorem or the smashing theorem, whose proofs have been read and understood by many of us. So there may be some room here to come up with a more conceptual approach to the disproof; then again, there may not.

  7. Classify all finite loop spaces. This is the long term project of Bill Dwyer and Clarence Wilkerson. The theory, I believe, is that the Lie groups are essentially the only examples. So one constructs Weyl groups and maximal tori and the like. But certainly at individual primes there can be other examples, like BD_3 at p=2.

  8. Say something general about the stable or unstable homotopy groups of spheres. For example, Ravenel has suggested that the size of the nth homotopy group of S^k grows polynomially in n, maybe even cubically. I presume he meant stable homotopy, but one could also ask the question unstably.

  9. The Kervaire invariant problem. This one is on the list not for its intrinsic importance (in my opinion!) but rather because so many, many people claim to have solved it, but all their proofs are wrong. The question is whether the element represented by h_i^2 in the Adams spectral sequence at the prime 2 is a permanent cycle or not. I am not sure what reference to give here, but Mark Mahowald is the world's expert on this problem and all related problems.

  10. Once again, I am not sure whether this problem deserves to be called major, but it is annoying that the the R. Cohen - Goerss result proving that h_0 h_i is a permanent cycle in the Adams spectral sequence for all primes bigger than 3 is wrong. The flaw was found by Minami, and it appears to be fatal to their proof. So this problem is still open.

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Mark Hovey
Department of Mathematics
Wesleyan University